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Strategic Financial Innovation in
Segmented Markets
By
Rohit Rahi
Jean-Pierre Zigrand
DISCUSSION PAPER NO 595
DISCUSSION PAPER SERIES
September 2007
Rohit Rahi obtained his Ph.D. in Economics from Stanford University in 1993. He is a
Reader in Finance and Economics and a member of the Financial Markets Group at the
London School of Economics. His research interests are in the theory of financial
markets, in particular arbitrage in segmented markets, security design, and asset
pricing with asymmetric information; and in general equilibrium theory with
incomplete markets. He has published articles in the Review of Economic Studies,
Journal of Economic Theory, and Journal of Business. He served a nine-year term on
the editorial board of the Review of Economic Studies. Jean-Pierre Zigrand is a
lecturer at the London School of Economics. His research interests are Asset Pricing,
Financial Intermediation Theory, and General Equilibrium Theory. Any opinions
expressed here are those of the authors and not necessarily those of the FMG. The
research findings reported in this paper are the result of the independent research of
the authors and do not necessarily reflect the views of the LSE.
Strategic Financial Innovation in
Segmented Markets∗
by
Rohit Rahi
Department of Finance,
Department of Economics,
and Financial Markets Group,
London School of Economics,
Houghton Street, London WC2A 2AE
and
Jean-Pierre Zigrand
Department of Finance
and Financial Markets Group,
London School of Economics,
Houghton Street, London WC2A 2AE
September 25, 2007.
∗
This paper has benefited from comments by Antoine Faure-Grimaud, Pete Kyle, Joel Peress and
especially Dimitri Vayanos. We also thank seminar participants at UC Berkeley, Cambridge, London Business School, Maastricht, Oxford, Stockholm School of Economics, the European Finance
Association Meetings, and the NBER/NSF General Equilibrium conference at UC Davis.
Abstract
We study a model with restricted investor participation in which strategic
arbitrageurs reap profits by exploiting mispricings across different market segments. We endogenize the asset structure as the outcome of a security design
game played by the arbitrageurs. The equilibrium asset structure depends realistically upon considerations such as depth and gains from trade. It is neither
complete nor socially optimal in general; the degree of inefficiency depends
upon the heterogeneity of investors.
Journal of Economic Literature classification numbers: G12, D52.
Keywords: Security design, arbitrage, intermediation, market segmentation.
2
1
Introduction
The optimal design of traded securities has been the subject of a growing body
of research. The focus in the literature has been on innovations carried out by
agents who do not themselves trade the securities they design, such as options or
futures exchanges, entrepreneurs who sell equity stakes in their firms, or abstract
social planners. In reality, agents involved in financial innovation are often profitseeking institutions that actively make markets and trade the new securities across
markets, for arbitrage or hedging purposes. A large chunk of their profits comes from
proprietary trading, and not simply from transaction fees received from investors.
The profits of the innovating agents arise from bid-ask spreads, as well as from price
differentials across markets or investor clienteles. Moreover, financial innovators are
typically not price-takers but large strategic institutions who know how their actions
affect prices. In this paper we propose a model that captures these features.
As an illustration, consider the following concrete examples. Trading opportunities exist between an exchange-traded fund (ETF) and the underlying portfolio
of shares (the “basket”). This arbitrage is one of the main motivations behind the
creation of ETFs by their sponsors, typically large brokers or specialists. The sponsors, also known as “authorized participants,” can create and redeem ETF shares
by swapping the basket for an ETF share and vice versa. For example, if the ETF
is trading at a discount relative to the basket, an authorized participant can deliver
an ETF share and receive the basket. Other investors cannot exploit this arbitrage
because they are excluded from the creation-redemption process. They must instead
employ a conventional long-short strategy which relies on mean-reversion in the discount over time. Moreover, discounts (and premia) are typically within the bid-ask
spread for ETF shares (Engle and Sarkar (2006)).
Similarly, profit opportunities may arise between derivatives exchanges and the
underlying markets. For instance, differential marginal valuations may lead to a
relative mispricing between the S&P 500 futures traded in Chicago and the basket of
underlying stocks traded in New York. By introducing exchange-traded derivatives,
exchanges open up such opportunities for their members. Even within an exchange,
such arbitrages are possible, either across related derivatives, or simply due to bidask spreads which capture the difference between the marginal valuations of buyers
and sellers. Anecdotal evidence suggests that, depending on markets and trading
conditions, a fair number of trades are matched directly, exposing market makers to
no risk.
Mispricings that are a source of arbitrageur profits are even more apparent in
over-the-counter (OTC) markets. For instance, consider the issuer of an arbitrage
collateralized debt obligation (CDO). In its cash form, the issuer purchases negotiable
assets, typically high-yield bonds, which it then uses as collateral for securitization.
The various tranches of the securitization are designed to suit the specific demands
of different clienteles. The optimal design of the tranches1 maximizes the arbitrage
1
Issuers are able to optimize the designed securities along several dimensions. A careful choice
3
profits of the issuer, which are equal to the difference between the price received
for the new securities and the price paid for the collateral. Only the issuer of the
CDO can exploit this arbitrage. Issuing a CDO is difficult and costly. It requires
first-class distribution capabilities, and typical upfront setup costs are in the region
of $5 million.
Or consider the new and fast-growing category of property derivatives known as
property total return swaps (TRSs). The intermediaries involved in these derivatives
are banks, specialized interdealer brokers and spread betting companies, often in collaboration with real estate brokers for their local knowledge. Typically, one party to
the swap, called the the total return payer, is endowed with an amount of physical
property (e.g. developers, shopping mall owners) and wishes to hedge against movements in the property market, while the other party, the total return receiver, wishes
to invest in the property market in order to diversify. The total return payer pays
the receiver the rate of return on a property index in exchange for a fixed or floating
interest rate on the notional amount of the swap. Thus the total return payer makes
a synthetic sale of property and the total return receiver makes a synthetic purchase
by paying interest. Direct investment in property, on the other hand, is difficult and
costly, with nonnegligible barriers to entry.2
One common thread underlying these examples is that securities are designed by
innovators who extract profits by exploiting differences in marginal investor valuations. The innovators desire as little downside risk as possible. In practice this leads
to financial innovations that are redundant to a certain extent, in the sense that they
can be satisfactorily replicated (or superreplicated) via a portfolio of the existing assets, at least by the most sophisticated and low-cost institutions. This raises the
question why investors who buy such an innovation, say a structured product sold at
a markup by an investment bank, do not replicate it themselves instead, and thereby
pocket the price differential. There are many reasons that come to mind—limited
knowledge regarding the right hedging strategy, high transaction costs, high setup
costs involved in buying a seat on an exchange or obtaining access to real-time data
and trading as required by delta-hedging, etc. In the case of ETFs and arbitrage
CDOs, for example, the arbitrage trade is effectively open only to the authorized participants and the issuers, respectively. In the TRS example, trading in the underlying
is costly or restricted, and price discovery is difficult.
In this paper, such impediments to perfect and costless replication are captured in
the assumption that various investor groups have restricted access to capital markets.
Their marginal valuations are therefore typically not equalized. The question we
would like to address then is the following: given markets with differential marginal
valuations, which securities are introduced by profit-maximizing innovators? And
of the attachment points of subordination, which defines the tranches, allows them to exploit the
analysis methods of the rating agencies, and thereby indirectly to select the clienteles to which the
tranches can be marketed. Issuers can also choose the collateral strategically; for instance a higher
degree of illiquidity would procure higher yields.
2
For instance, acquisition costs of property in the UK are around 8%, and in many countries
the costs are higher still. Some jurisdictions outright disallow foreigners from purchasing property.
4
what are the welfare properties of these innovations?
That trading occurs locally and might translate into exploitable arbitrage opportunities globally has been known for many centuries of course. Postlethwayt (1757)
provides a fascinating account of the opportunities for arbitrage profits in the exchange network of Europe connecting London, Amsterdam, Paris and a dozen other
cities in the 17th and 18th centuries. In present-day markets, the importance of
financial innovation originating with intermediaries that facilitate risk-sharing for
agents who find it costly to trade directly with each other has been documented by
Allen and Santomero (1997). They also point to the absence of a theoretical framework to address this feature of financial markets. The present paper takes a first
step in filling this gap.
We study a two-period model with asset trading at date zero and uncertainty
resolved at date one. There are several market segments or “exchanges.” With each
exchange is associated a group of competitive investors, who for simplicity conform to
a version of the CAPM. Investors may only trade the assets available on their “local”
exchange. As a special case they may constitute a homogeneous clientele that does
not trade within itself. In addition, there are agents who are “global” players—they
are able to trade on all exchanges simultaneously. These agents profit by arbitraging
away price differentials across exchanges. We refer to them as “arbitrageurs.” They
have zero initial wealth, so they can be interpreted as pure intermediaries. Any
transfer of resources across exchanges is intermediated by the arbitrageurs.
We first solve for equilibrium for a given asset structure. This asset structure
may be completely arbitrary, with the assets trading on one exchange bearing no
specific relationship to those trading on another. To any amount of asset supplies
by arbitrageurs to the exchanges, there corresponds a Walrasian equilibrium on each
exchange. Equilibrium supplies are then determined in a Cournot game played by
the arbitrageurs. The result is a unique Cournot-Walras equilibrium associated with
each asset structure.
We then endogenize the asset structure as the outcome of a security design game
among the arbitrageurs before any trade takes place. Arbitrageurs determine the
asset structure on any given exchange by adding assets available for trade (not necessarily the same set across exchanges). In the subsequent trading game, all arbitrageurs can trade any of the securities that have been introduced, while investors
can trade the securities introduced on their own exchange. An arbitrageur’s payoff in the security design game is his trading profit in the ensuing Cournot-Walras
equilibrium. The arbitrageurs can thus be viewed as intermediaries who can target
their clients according to their needs and supply them with securities that were hitherto unavailable to them but may be globally redundant. The financial innovations
together with the inter-market trades can be viewed as a means of integrating the
various markets.
We show that there is a unique equilibrium of the security design game in which
there is a single asset on each exchange. In the case in which there are only two
exchanges, this asset is the difference between the autarky (absent arbitrageur activ5
ity) state-price deflators of these exchanges. In the case of multiple exchanges, the
equilibrium asset on an exchange is the difference between its autarky state-price
deflator and a weighted sum of the autarky state-price deflators of all exchanges.
This weighted sum is in fact the complete-markets Walrasian state-price deflator
of the entire integrated economy. The intuition for this result is that by buying
their own state-price deflator, investors on a given exchange effectively sell their
exchange-specific idiosyncratic aggregate endowment, and by simultaneously selling
the Walrasian state-price deflator they buy a fraction of the overall aggregate endowment, thereby diversifying their risk. Since diversification is optimal for investors,
arbitrageurs extract the maximal amount of profits by offering this economical and
desired security structure. If we view the various exchanges as representing countries,
we find a version of Shiller’s “macro markets” (Shiller (1993)).
The equilibrium security design is optimal for arbitrageurs in the sense that no
other asset structure yields higher arbitrageur profits. Furthermore, if investors on
each exchange are identical, but possibly heterogeneous across exchanges, the equilibrium asset structure is actually Pareto optimal (though the equilibrium allocation
is not, since arbitrageurs are imperfectly competitive). Relative to an arbitrary initial asset structure, however, equilibrium innovation by arbitrageurs may hurt some
investors. We characterize who wins and who loses, and provide sufficient conditions
for all investors to gain. Finally, we note that if there are heterogeneous investors
within exchanges, the equilibrium security design fails to be Pareto optimal, since
arbitrageurs profit only from trade between exchanges and not from trade within exchanges. They might therefore not offer the precise assets that would allow investors
to exhaust gains from trade within an exchange.
One contribution of our paper is to endogenously derive an asset structure which
is incomplete, without imposing a bound on the number of assets that may be introduced. Moreover, the assets that arbitrageurs innovate in our model may be
redundant from the economy-wide perspective. This is an aspect of actual financial innovation that has often been remarked on in the literature, but cannot be
accounted for by previous research that has focused for the most part on frictionless
environments. Beyond the security design results, our model provides an explicit
characterization of intermediation across segmented markets with an arbitrary asset
structure. These intermediaries may be interpreted as agents traditionally thought
of as arbitrageurs, such as hedge funds or proprietary trading desks of investment
banks, or as market makers trading a given set of securities.
The paper is organized as follows. We introduce the framework and notation
in Section 2. In Section 3 we solve for the equilibrium of the trading game for
an arbitrary asset structure. Still maintaining an exogenous asset structure, we
investigate the role of arbitrageurs in integrating markets in Section 4. Our security
design and welfare results are in Sections 5 and 6 respectively. We relate macro
markets to our setup in Section 7. In Section 8 we review the literature, in particular
the theoretical research on security design and the empirical evidence on segmented
markets. Section 9 concludes. Proofs of results in the main text are collected in the
6
Appendix.
2
The Setup
We consider a two-period economy in which assets are traded at date 0 and pay off
at date 1. Assets are traded in several locations or “exchanges.” They are in zero
net supply. We do not impose complete markets or the existence of a riskless asset.
Investor i ∈ I k := {1, . . . , I k } on exchange k ∈ K := {1, . . . , K} has endowments
k,i
(ω0 , ω k,i ), where ω0k,i ∈ R is his endowment at date 0, and ω k,i , a (real-valued)
random variable, is his endowment at date 1. His preferences are given by quasilinear
quadratic expected utility
1 k,i k,i 2
k,i
k,i k,i
k,i
k,i
U (x0 , x ) = x0 + E x − β (x ) ,
2
k,i
where xk,i
is a random variable representing
0 ∈ R is consumption at date 0, and x
k,i
consumption at date 1. The coefficient β is positive. Investors are price-taking
and can trade only on their own exchange. It will be useful later
P tok,icharacterize
k
−1 −1
exchange k in terms of its aggregate preference
parameter
β
:=
[
i (β ) ] , and
P
k,i
its aggregate date 1 endowment ω k := i ω
we defineP
the corresponding
P . Similarly
k −1 −1
parameters for the entire economy: β := [ k (β ) ] and ω := k ω k . Due to the
non-monotonicity of quadratic utility, we need to assume that 1 − βω ≥ 0. It says
that the representative investor with aggregate preference parameter β is weakly
nonsatiated (has non-negative marginal utility) at the aggregate endowment ω.
In addition to investors there are N arbitrageurs, with typical arbitrageur n ∈
N := {1, . . . , N }, who possess the technology to trade both within and across exchanges. Arbitrageurs are imperfectly competitive. They have no endowments. For
simplicity, we assume that they only care about date 0 consumption.
There are J k assets on exchange k, with typical asset j paying off a random
quantity dkj at date 1. Assets on exchange k can then be represented by the random
payoff vector dk := (dk1 , . . . , dkJ k ). We assume that there are no redundant assets on
exchange k. We also assume that all assets are arbitraged, i.e. traded by both the
local investors and the arbitrageurs.3 The economy-wide asset structure {dk }k∈K is
endogenously determined as described below.
We assume that all random variables have finite support. Then we can describe
the uncertainty by finitely many states of the world s ∈ S := {1, . . . , S}.
While the law of one price may not hold across exchanges, in equilibrium it must
hold within any exchange. This is equivalent to the existence of a state-price deflator, one for each exchange. Given an asset price vector q and asset payoff vector
3
This is an innocuous assumption. It is straightforward to extend our analysis to the case where,
on a given exchange, some assets are not arbitraged, i.e. traded only by investors on the exchange,
while other assets are arbitraged. It turns out, however, that equilibrium prices of arbitraged assets
are not affected by the payoffs of non-arbitraged assets. Thus the characteristics of non-arbitraged
assets have no bearing on arbitrage trades or on security design by arbitrageurs.
7
d, a random variable p is called a state-price deflator if qj = E[dj p] for every asset
j, or more compactly, q = E[dp]. In the literature, the term “state-price deflator”
is often used interchangeably with the terms “state-price density,” “stochastic discount factor,” or “pricing kernel.” Much of the intuition of the present paper can
be gathered from comparing state-price deflators. Since arbitrageurs are strategic,
they know that their choice of asset payoffs and asset supplies affects equilibrium
state-price deflators.
We model the activities of arbitrageurs as a subgame-perfect Nash equilibrium of
a two-stage game. In the first stage arbitrageurs design securities, resulting in some
asset structure {dk }k∈K . In the second stage they trade these securities. We solve
the game backwards, starting with the second stage. We specify the details of the
game in each stage when we come to it.
3
Cournot-Walras Equilibrium
In this section we analyze the trading game for exogenously given asset payoffs.
Let
P
k,n
k
y be the supply of assets on exchange k by arbitrageur n, and y := n∈N y k,n
the aggregate arbitrageur supply on exchange k. For given y k , q k (y k ) is the marketclearing asset price vector on exchange k, with the asset demand
Pn of investor i on
k,i k
n
exchange k denoted by θ (q ). For vectors v, w ∈ R , v · w := i=1 vi wi .
Definition 1 Given an asset structure {dk }k∈K , a Cournot-Walras equilibrium (CWE)
of the economy is an array of asset price functions, asset demand functions, and ark
k
k
k
k
bitrageur supplies, {q k : RJ → RJ , θk,i : RJ → RJ , y k,n ∈ RJ }k∈K, i∈I k , n∈N , such
that
1. Investor optimization: For given q k , θk,i (q k ) solves
h
β k,i k,i 2 i
k,i
(x )
max xk,i
+
E
x
−
0
2
θk,i ∈RJ k
subject to the budget constraints:
k,i
k
k,i
xk,i
0 = ω0 − q · θ
xk,i = ω k,i + dk · θk,i .
0
2. Arbitrageur optimization: For given {q k (y k ), {y k,n }n0 6=n }k∈K , y k,n solves
X
X
0
y k,n · q k y k,n +
y k,n
max
y k,n ∈RJ k k∈K
s.t.
n0 6=n
X
k∈K
8
dk · y k,n ≤ 0.
3. Market clearing:
q k (y k )
X
k∈K
solves
θk,i (q k (y k )) = y k ,
∀k ∈ K.
i∈I k
Note that investors take asset prices as given, while arbitrageurs compete Cournotstyle. This equilibrium concept is due to Gabszewicz and Vial (1972), and a review
can be found in Mas-Colell (1982). Arbitrageurs maximize time 0 consumption, i.e.
profits from their arbitrage trades, but subject to the restriction that they are not
allowed to default in any state at date 1. Equivalently, arbitrageurs need to be
completely collateralized.
We first solve for Walrasian equilibrium on exchange k for a given vector of asset
supplies y k . It is convenient to think of valuation in terms of state-price deflators:4
Lemma 3.1 (Market-clearing prices for given arbitrageur supplies) For given
arbitrageur supply y k , the following is a state-price deflator for exchange k:
p̂k (y k ) := pk − β k (dk · y k ),
(1)
where pk := 1 − β k ω k .
The function p̂k (y k ) is the inverse demand function that arbitrageurs face when
supplying state-contingent consumption dk · y k to exchange k.5 The parameter β k
measures the “depth” of exchange k: it is the price impact of a unit of arbitrageur
trading.6 For instance, ceteris paribus, the market impact of a trade is smaller on
exchanges with a larger population; it can be absorbed by more investors.
Using (1), and denoting by xk := ω k + dk · y k the aggregate date 1 consumption
on exchange k, asset prices are given by
qjk (y k ) = E[dkj p̂k (y k )] = E[dkj (1 − β k xk )] =
E[dkj ]
| {z }
risk neutral price
−
β k E[dkj xk ]
| {z }
risk aversion discount
Thus we can interpret the price of asset j on exchange k as the risk neutral price
from which a risk aversion discount is subtracted. The discount depends on the risk
aversion parameter β k as well as on the diversification benefits of asset j, as measured
by E[dkj xk ] = cov[dkj , xk ] + E[dkj ]E[xk ]. Ceteris paribus, the more an asset covaries
with aggregate consumption, the lower its price must be in equilibrium.
4
If markets are incomplete on exchange k, there is a multiplicity of state-price deflators p for
given (q k , dk ), all of which satisfy q k = E[dk p]. We will often find it convenient to choose a particular
state-price deflator—it should be kept in mind that this imposes no restriction as far as asset prices
are concerned.
5
That p̂k is affine in y k is a direct consequence of our assumption of quasilinear quadratic
preferences. As is well known in Cournot theory, Cournot equilibria can only be guaranteed if
inverse demand functions are affine in supplies.
6
More precisely, the state s value of the state-price deflator falls by β k for a unit increase in
arbitrageur supply of s-contingent consumption.
9
We say that exchange k is in autarky if y k = 0. It is clear from (1) that pk is
exchange k’s autarky state-price deflator. The deflators {pk }k∈K will play a key role
in this paper. Analogous to pk , pk,i := 1 − β k,i ω k,i is the no-trade state-price deflator
of investor (k, i), meaning that the investor chooses θk,i = 0 when faced with the
deflator pk,i .7
Our assumptions on preferences, in conjunction with the absence of nonnegativity
constraints on consumption, guarantee that the equilibrium pricing function on an
exchange does not depend on the initial distribution of endowments, but merely on
the aggregate endowment of the local investors. The autarky state-price deflator pk
also does not depend on dk , even though investors on exchange k do trade these
assets among themselves.
Before proceeding further we need to introduce some additional notation. In
general, markets are incomplete on exchange k. Hence a random variable x may not
k
be marketable, meaning that no portfolio has x as its payoff: there is no θ ∈ RJ for
which x = dk ·θ. The set of marketable payoffs M k is the set of all portfolio payoffs on
exchange k; alternatively it is the span of dk , denoted by hdk i. Thus M k := hdk i :=
k
{x : x = dk · θ, for some portfolio θ ∈ RJ }. Given the linear space M k , a random
variable x can be split into a marketable component xM k and a non-marketable
component in such a way that the mean-square distance between x and xM k is
minimal. This marketable component xM k is given by the least-squares regression of
x on dk . In other words, there is a unique decomposition x = xM k +, with E[dk ] = 0,
and xM k the payoff of a portfolio of the assets dk , such that E[(xM k − x)2 ] is minimal.
We now solve the Cournot game among the arbitrageurs, given the inverse demand functions (1) for each k ∈ K. It turns out that there is a unique CWE, and
that this equilibrium is symmetric, i.e. y k,n is the same for all n. It is convenient to
state arbitrageur supplies in terms of the supply of state-contingent consumption:
Lemma 3.2 (Equilibrium supplies) Equilibrium arbitrageur supplies are unique
and symmetric. For asset structure {dk }k∈K , they are given by
dk · y k,n =
1
k
A
p
k − pM k ,
M
(1 + N )β k
k∈K
(2)
where pA ≥ 0 is a state-price deflator for the arbitrageurs.
The random variable pA is a state-price deflator for the arbitrageurs in the sense
that it is a state-price deflator, i.e. q k = E[dk pA ] for all k, and moreover pA (s)
is the arbitrageurs’ marginal shadow value of consumption in state s (formally, as
shown in the proof of Lemma 3.2, pA (s) is the Lagrange multiplier attached to the
arbitrageurs’ no-default constraint in state s). Assuming for the moment that pk
and pA are marketable, we can clearly see from (2) that arbitrageurs supply state
s consumption to exchange k when the price that agents on k are willing to pay,
7
This can be seen directly from equation (12) in the proof of Lemma 3.1 in the Appendix.
10
pk (s), exceeds the arbitrageurs’ own valuation, pA (s).8 This statement should be
qualified, since while these are the “optimal” supplies in some sense (to be confirmed
subsequently), they may not be marketable, i.e. they may not be implementable
via a portfolio of the existing securities. Therefore, arbitrageurs will supply state s
consumption to exchange k if the marketable component of the excess willingness to
pay, (pk − pA )M k = pkM k − pA
, is positive in state s.
Mk
The factor of proportionality in (2) is determined by two considerations. First,
the deeper is exchange k (i.e. the lower is β k ), the more arbitrageur n trades on this
exchange, since he can afford to augment his supply without affecting margins as
much. And second, the supply vector is scaled to zero as competition intensifies,
because the whole pie shrinks and there are more players to share the smaller pie
with (see also Lemma 3.4 below).
Note that generically all arbitrageurs trade across all exchanges (provided there
are arbitrage opportunities). In particular, arbitrageurs never carve up the set of
exchanges amongst themselves with a view to reducing competition.
Lemma 3.2 gives us the equilibrium supply y k,n of arbitrageur n. The total
equilibrium supply is then y k = N y k,n . Substituting into the pricing equation (1)
determines the equilibrium prices:
Lemma 3.3 (Equilibrium prices) The following is an equilibrium state-price deflator for exchange k:
1
N
pk +
pA .
(3)
p̂k =
1+N
1+N
In particular, as N goes to infinity, the equilibrium valuation on each exchange converges to the arbitrageurs’ valuation: limN →∞ p̂k = pA .
Intuitively, in equilibrium, prices on exchange k reflect in a convex fashion both the
marginal valuation of the investors on k after having exhausted all gains from trade
amongst themselves, pk , as well as the marginal valuation of the arbitrageurs, pA .
For N = 0, equilibrium state prices are equal to autarky state prices: p̂k = pk . As
N → ∞, the equilibrium state-price deflator converges to the arbitrageur’s marginal
valuation. The arbitrageurs’ valuation in turn does not depend on k, but rather
depends globally on the marginal valuations of all exchanges.
Finally, we calculate the equilibrium profits
n (which do not depend
P kof arbitrageur
k,n
on n since the CWE is symmetric), Φ := k q · y .
Lemma 3.4 (Equilibrium profits) The equilibrium profits of an arbitrageur, for
given asset structure {dk }k∈K , are given by
X 1
1
2
·
E[(pkM k − pA
(4)
Φ({dk }) =
M k ) ].
2
k
(1 + N )
β
k
8
Assume for instance that there are three exchanges, with complete markets on each exchange,
and that p1 (s) < p2 (s) < p3 (s). We will show later (Lemmas 4.1 and 4.2) that pA is a convex
combination of the pk ’s. Then, from Lemma 3.2, arbitrageurs necessarily buy state s consumption
on exchange 1 and sell on exchange 3. They may buy or sell on exchange 2; they buy on 2 if and
only if p2 (s) < pA (s).
11
A typical arbitrageur’s equilibrium profits are equal to the sum across exchanges
of the size of (the marketable component of) the difference between the marginal
investors’ willingness to pay in autarky and the arbitrageur’s shadow value, scaled
down by shallowness and by the degree of competition. As N goes to infinity, individual arbitrageur trades vanish, as do total arbitrageur profits N Φ. The vanishing
of aggregate arbitrageur profits as N increases without bound suggests that the arbitrageurs perform the duties of a Walrasian auctioneer in the limiting equilibrium.
4
A Walrasian Benchmark
It turns out that the equilibrium of the arbitraged economy that we have just computed bears a close relationship to an appropriately defined competitive equilibrium,
with no arbitrageurs. This relationship is somewhat subtle, however. The reader
is referred to Rahi and Zigrand (2007b) for an analysis of the general case. In this
section we restrict ourselves to asset structures that satisfy a certain spanning condition which holds at the equilibrium security design and also suffices for our welfare
results. Under this spanning condition, we show that arbitrageur valuations and
Walrasian valuations coincide.
Let p∗ denote the complete-markets Walrasian state-price deflator of the entire
integrated economy with no participation constraints. We have:
Lemma 4.1 (Walrasian benchmark) The complete-markets Walrasian state-price
deflator p∗ is given by
X
λk pk ,
p∗ =
k∈K
where
k
λ :=
1
βk
PK 1
j=1 β j
,
k∈K
and the complete-markets Walrasian net trades between exchanges are
1 k
(p − p∗ ),
βk
k ∈ K.
(5)
Furthermore, p∗ = 1 − βω ≥ 0.
The state-price deflator p∗ reflects the investors’ economy-wide average willingness
to pay, with the willingness to pay on each exchange weighted by its relative depth.
We can now state the spanning condition alluded to above:
(S) Either (a) dk = d, k ∈ K, or (b) pk − p∗ ∈ M k , k ∈ K.
Under S(a) we have an standard incomplete markets economy in which all investors
can trade the same set of assets. S(b) is the condition that characterizes an equilibrium security design, as we shall see in the next section.
12
Lemma 4.2 (Arbitrageur valuations) Under S, arbitrageur valuations in the CWE
coincide with valuations in the complete-markets Walrasian equilibrium, i.e. we can
choose pA = p∗ .
We showed in Lemma 3.3 that the state-price deflator p̂k converges to pA as the
number of arbitrageurs grows without bound. Hence we have:
Proposition 4.1 (Convergence to Walrasian equilibrium) Suppose the asset
structure satisfies condition S. Then the equilibrium valuation on exchange k in the
arbitraged economy converges to the complete-markets Walrasian equilibrium valuation as the number of arbitrageurs N goes to infinity, i.e. limN →∞ p̂k = p∗ .
Thus asset prices in the arbitraged economy converge to asset prices in the completemarkets Walrasian equilibrium: limN →∞ q k = E[dk p∗ ]. It is in this sense that arbitrageurs serve to integrate markets.
It should be noted that, as N goes to infinity, we get convergence to the completemarkets Walrasian equilibrium valuation (under S). In general, however, we do not
get convergence to the complete-markets Walrasian equilibrium allocation.9 A sufficient condition for the latter is complete markets on each exchange. As we shall
see later (Proposition 6.1), S(b) suffices as well if investors are identical within each
exchange.
Proposition 4.1 can be viewed as a formal statement of the often-heard comment
that the current proliferation of arbitrage hedge funds leads to more efficient markets.
However, in view of Lemma 3.4, these funds cannot be viewed as an asset class by
themselves (like equities and bonds, for example) since a higher number of funds
leads to lower fund returns, both individually and in the aggregate, converging to
zero in the limit.
5
Security Design by Arbitrageurs
We have seen that there is a unique CWE associated with any asset structure
{dk }k∈K . In this section we endogenize the security payoffs.
Arbitrageurs play the following security design game. Each arbitrageur introduces
securities on each exchange, which are then traded by all arbitrageurs. Formally, arbitrageur n chooses security payoffs {dk,n }k∈K . Given a strategy profile {dk,n }k∈K,n∈N ,
the asset structure for the economy is {dk }k∈K , where dk = ∪n∈N dk,n (here union
has the obvious meaning: an asset is in dk if and only if it is in dk,n , for some n).
The payoffs of arbitrageurs are the profits they earn in the CWE associated with
this asset structure. Due to symmetry of the trading game, all arbitrageurs have
the same payoff. We say that an asset structure {dk }k∈K is a Nash equilibrium of
the security design game if dk = ∪n∈N dk,n , all k, for some Nash equilibrium profile
9
Under S, the allocation converges to the restricted-participation Walrasian equilibrium allocation; see Rahi and Zigrand (2007b) for details.
13
{dk,n }k∈K,n∈N . An asset structure is optimal for arbitrageurs if it yields the highest
payoff for arbitrageurs, among all possible asset structures.
Proposition 5.1 The following statements are equivalent:
1. pk − p∗ ∈ hdk i, for all k ∈ K;
2. the asset structure {dk }k∈K is optimal for arbitrageurs;
3. the asset structure {dk }k∈K is a Nash equilibrium of the security design game.
Thus the complete asset structure, with complete markets on every exchange,
is optimal for arbitrageurs, and a Nash equilibrium. Given the complexity of the
real world, this would require a very large number of securities. However, all optimal/equilibrium asset structures are payoff-equivalent for arbitrageurs. Among
these, a minimal asset structure is one with the smallest number of assets. Such an
asset structure would be the one chosen if each security issued bore a fixed cost c, no
matter how small.10 Our main security design result is an immediate consequence of
Proposition 5.1:11
Proposition 5.2 (Security design) The asset structure
dk = pk − p∗ ,
k∈K
is
1. the unique minimal optimal asset structure for arbitrageurs; and
2. the unique minimal Nash equilibrium of the security design game.
The optimal asset structure for arbitrageurs spans the net trades between exchanges in the complete-markets Walrasian equilibrium, which are given by (5). The
minimal optimal security, and the unique minimal equilibrium, on exchange k is a
swap, exchanging the autarky state-price deflator of exchange k for the completemarkets Walrasian state-price deflator of the entire integrated economy. If there are
only two exchanges, say 1 and 2, then the optimal securities p1 − p∗ and p2 − p∗ are
both proportional to p1 − p2 , the difference of the autarky state-price deflators of the
two exchanges.
A reading of the optimal arbitrage supply dk · y k,n , which is proportional to
pkM k − pA
(from Lemma 3.2), suggests that the optimal security design for arbiMk
trageurs should be as given in the proposition. When arbitrageurs must operate with
exogenously given assets, they supply state-contingent consumption to exchange k
that is (up to a scalar multiple) as close as possible to pk − pA using the given assets,
i.e. pkM k − pA
. If now they can design securities, they can supply exactly pk − pA
Mk
10
11
In fact, such fixed costs are significant; see Tufano (1989) for an empirical assessment.
When we say “unique”, we mean “unique up to scaling.”
14
rather than the best approximation of pk − pA . Moreover, under this security design,
pA = p∗ from Lemma 4.2. A notable feature of the equilibrium asset structure is
that it allows arbitrageurs to run what practitioners call a “matched book,” i.e. an
asset position with exactly offsetting future payoffs.
Note that a single security on each exchange suffices for arbitrageurs to maximize
their profits, and our result therefore generates incomplete markets endogenously,
without any constraint on the number of securities. The reason is that, within any
exchange k, asset prices are determined by a Walrasian auction and arbitrageurs do
not profit from those intra-exchange trades. Intuitively, arbitrageurs only profit from
mispricings between the market price of the innovation and the replicating portfolio.
They are therefore only concerned with the one-dimensional net trade they mediate
between k and the rest of the economy, which can be accomplished via a single
security collinear with the desired net trade. Relatedly, if pk = p∗ , we claim that
arbitrageurs do not find it profitable to introduce any assets on exchange k. But isn’t
it true that, with heterogeneous investors on exchange k, there must be some agent
willing to pay or receive an amount different from the one on some other exchange,
and therefore provide an incentive to innovate? The answer is no, since if such an
asset is sold to investor (k, i), all other agents on k can trade that same asset as
well, by nonexclusivity. The resulting price established on exchange k is such that
no arbitrage opportunities arise across exchanges: pk = p∗ .
Finally, it is interesting to realize that although net trades, and therefore equilibrium allocations, depend on the degree of competition N , the equilibrium asset
structure does not depend on N . This is a feature of the linear-quadratic model in
which demand functions are linear, and depth is a constant independent of trading
volume. We discuss this further in the next section.
Our analysis readily extends to the more realistic case where arbitrageurs can
innovate on all exchanges, but there are pre-existing assets whose payoffs {dk } they
cannot affect. In other words, security design really represents incremental innovation. For instance, if there is some exchange k which has complete markets, one can
interpret the result as the optimal design of redundant derivative securities on the
other exchanges. The following result is immediate from Proposition 5.1.
Proposition 5.3 (Innovation) For given {dk }k∈K , the asset structure
[dk
(pk − p∗ )]
dk
if
pk − p∗ 6∈ hdk i,
if
pk − p∗ ∈ hdk i,
is
1. a minimal optimal asset structure for arbitrageurs; and
2. a minimal Nash equilibrium of the security design game.
Since for a given {dk } arbitrageurs find it optimal to supply state-contingent
consumption proportional to pk − p∗ if allowed, they innovate on exchange k if and
15
only if pk − p∗ 6∈ hdk i, in which case they “complete” the market by adding pk − p∗ .
Equivalently, they could add a security that makes pk − p∗ tradable in conjunction
with dk . For this reason, we can no longer say that [dk (pk − p∗ )] is the unique
minimal asset structure.
Example 1 (Property Total Return Swaps) Recall, from our discussion in the
Introduction, that a TRS swaps the rate of return on a property index with a prespecified interest rate. We show how our framework can generate a TRS as an
equilibrium security.
Assume that in terms of property exposure the market can be split into K ≥ 2
clienteles. Clientele 1 represents investors who are endowed with a given type of
commercial property ω̂. The remaining clienteles initially have no exposure to such
commercial property but would like to diversify into it. Let us also assume for
simplicity that all clienteles are reasonably diversified with respect to all other sectors,
ω̌, with clientele k owning a fraction φk of the global aggregate holdings. We can
view the non-property sector to be a global equity index for concreteness. Then
the endowments are ω 1 = ω̂ + φ1 ω̌ and ω k = φk ω̌,P
k ≥ 2. The (not necessarily
k
marketable) global market portfolio is therefore ω = K
k=1 ω = ω̂ + ω̌.
∗
Using the fact that p = 1 − βω, the equilibrium security on exchange 1 is
1
p − p∗ = (β − β 1 )ω̂ + (β − β 1 φ1 )ω̌. Similarly, the equilibrium security introduced on
exchange k 6= 1 is pk − p∗ = β ω̂ + (β − β k φk )ω̌. If all clienteles already have access
to the global equity index through their local exchanges (ω̌ ∈ hdk i, all k ∈ K), then,
from Proposition 5.3, the same security ω̂ marketed to each clientele is a minimal
equilibrium security design.
This security is simply a claim to the property ω̂. Let us denote its equilibrium
date 0 price by q. If there is a global risk free asset with rate r, the equilibrium
security can equivalently be represented by the time 1 payoff ω̂ − (1 + r)q. This
is exactly the payoff of a TRS. Just as in the real world, the contract swaps the
economic interest in the underlying property but allows clientele 1 to remain the
owner and to continue to enjoy the non-economic convenience yield of the property,
if any.
Example 2 (Survivor Swaps, a.k.a. Mortality Swaps) Defined benefit pension
plans and annuity providers are naturally long longevity risk, while life insurers, pharmaceutical companies and long term care homes are naturally short longevity risk.
Survivor swaps (as described for instance in Dowd et al. (2006)) are intermediated
by banks, in which the preset-rate leg is linked to a published mortality projection,
and the floating leg is linked to realized mortality. The attractions of these arrangements are the obvious ones of risk mitigation and capital release for those laying off
mortality risk, and good risk diversification due to low comovement with the overall
market for those taking it on.
16
6
Security Design and Social Welfare
For each asset structure {dk }k∈K , there is a unique CWE with the corresponding
equilibrium payoffs for each arbitrageur and investor. In this section we do a welfare
comparison of alternative asset structures by comparing the equilibrium payoffs associated with them. We say that an asset structure is socially optimal if it is Pareto
optimal for the set of all agents, arbitrageurs and investors.
Equilibrium arbitrageur profits are given by Lemma 3.4. Equilibrium utilities of
investors are as follows:
Lemma 6.1 (Equilibrium utilities) The equilibrium utility of investor (k, i), for
given asset structure {dk }k∈K , is (an affine function of )
"
2 #
N
1
k,i
.
(pk k − pA
W k,i := β k,i E[(dk · θk,i )2 ] = k,i E
(pM k − pkM k ) +
Mk
β
1+N M
Note that W k,i is a particular affine transformation of U k,i , being equal to zero
when the agent does not trade. Quite intuitively, the utility gains from trade are
higher (roughly speaking) the greater is the difference between an investor’s no-trade
valuation and the autarky valuation of his exchange, on the one hand, and the greater
is the difference between the autarky valuation of his exchange and the economy-wide
valuation pA , on the other.
We say that investors on exchange k are homogeneous if they have the same
no-trade valuations, i.e. pk,i = pk , for all i ∈ I k . We refer to an economy in which
investors are homogeneous within each exchange as a clientele economy. From the
point of view of arbitrageurs, each clientele k ∈ K consists of agents with identical
characteristics.
We will focus now on a clientele economy, returning to the general heterogeneous
agent case at the end of the section. Lemma 6.1 gives us the following welfare index
for clientele k:
2
X
1
N
k
k,i
2
W :=
W = k
E (pkM k − pA
(6)
Mk ) .
β
1
+
N
k
i∈I
P
Comparing this to (4), we see that the egalitarian social welfare function k W k
is proportional to arbitrageur profits Φ. Hence an asset structure that maximizes
arbitrageur profits is in fact socially optimal:12
12
This result has been foreshadowed by Postlethwayt (1757) who states in his entry on arbitrage
that “It does not always fall out, that the interest of private traders coincides with that of the
nation in general; but in the present case, it does: for while our merchants of ingenuity are gaining
advantages by themselves, by their skills in the exchanges, they necessarily contribute to rule and
control the courses of exchange in general, more and more in favor of our country than otherwise
they could be.”
17
Proposition 6.1 (Optimality: clientele economy) In a clientele economy, an
optimal asset structure for arbitrageurs (which is also a Nash equilibrium of the
security design game) is socially optimal.
This is not surprising given that the optimal arbitrageur-chosen securities span
the net trades between exchanges in the complete-markets Walrasian equilibrium of
the integrated economy. Consider the minimal equilibrium security design {pk − p∗ }.
The intuitive
reason why pk − p∗ is the right asset for exchange k is diversification.
P
Let θk := i∈I k θk,i be the aggregate portfolio of exchange k, which in equilibrium is
equal to the aggregate arbitrageur supply y k . From Lemma 3.2, exchange k receives
state-contingent consumption equal to
N
(pk − p∗ )
(1 + N )β k
β
N
k
=
ω−ω
1 + N βk
dk · y k =
(7)
where we recall that pk := 1−β k ω k , and p∗ = 1−βω by Lemma 4.1. Thus investors on
exchange k get rid of their idiosyncratic risk ω k and acquire a proportional position
in the global undiversifiable market portfolio ω, with a constant of proportionality
that depends on their relative level of risk tolerance.
Another way to see why the security pk − p∗ is optimal for clientele
P k is the
following. From Lemma 3.3, pk − p∗ is collinear with p̂k − p̂λ , where p̂λ := j∈K λj p̂j .
In general, it is well-known (see Magill and Quinzii (1996)) that the state-price
deflator evaluated at an equilibrium is locally the most valued security for an agent.
In our case, this is p̂k for clientele k. The optimal security for clientele k allows agents
in this group to obtain the payoffs of their most valued security p̂k , while shorting
the payoffs of the most valued securities of other clienteles {p̂j }j6=k . In equilibrium,
agents are induced to hold the swap by prices and by the underlying motivation to
diversify. It is a consequence of our linear-quadratic formulation that the optimal
asset structure at the arbitraged equilibrium, namely {p̂k − p̂λ }, is the same as the
optimal asset structure at the autarky equilibrium, {pk − p∗ }. Thus the optimal
security design in the arbitraged economy depends only on the autarky equilibrium
and not on the amount supplied by arbitrageurs.
However, while the equilibrium securities correspond to socially desirable ones,
the allocation that results is not Pareto optimal; it is merely Pareto-undominated
within the class of CWE allocations for different asset structures. Arbitrageurs are
strategic and restrict their asset supplies in order to benefit from the markup. This
implies that not all gains from trade are exhausted. We can see this by comparing the
net trades in the complete-markets Walrasian equilibrium, given by (5), and the net
trades at the CWE, given by (7). The latter trades are lower by a factor N/(1 + N )
due to imperfect competition. For N = 0, no consumption can be intermediated since
there are no intermediaries. More diversification is achieved as N increases, but it
is only when N tends to infinity that equilibrium allocations converge to Walrasian
allocations, and therefore to a Pareto optimum.
18
A socially optimal asset structure is not necessarily optimal for each clientele.
For example, starting from an initial asset structure {dk }, if markets are completed
for each clientele, the resulting security design is socially optimal. However, some
clienteles may be worse off since prices are typically affected by the introduction of
new securities and lead to redistributions. This possibility has been discussed by Elul
(1999), for instance. The following proposition addresses the important question as
to who gains and who loses as a result of an optimal financial innovation, and what
the drivers are.
Proposition 6.2 (Welfare gains and losses) In a clientele economy with initial
asset structure {dk }k∈K , clientele k is worse off at a socially optimal asset structure
if and only if
2
E[(pk − p∗ )2 ] < E[(pkM k − pA
M k ) ].
This follows directly from (6). Clientele k is worse off if and only if the total marketable gains from trade are smaller after the innovation than before.
Example 3 Consider an economy consisting of three exchanges, 1, 2 and 3, with
a single agent on each exchange. There are two states, S = {1, 2}, with π1 and
π2 the probabilities of states 1 and 2 respectively. Asset payoffs are as follows.
Exchange 1 has one security paying off 1 if state 1 obtains, and nothing if state 2
obtains. Exchange 2 trades a riskless bond paying 1 in both states. Exchange 3 has
a complete set of Arrow securities, Arrow security s paying 1 if and only if state s
occurs. Exchanges 1 and 2 are equally deep, with β 1 and β 2 both equal to β̄, which
satisfies
π1
.
(8)
0 < β̄ <
1 + π1
Date one endowments are deterministic and given by ω 1 = 1, ω 2 = 1/β̄ − 1 and
ω 3 = 1/(2β 3 ). Autarky state-price deflators are, therefore, p1 = 1 − β̄, p2 = β̄ and
p3 = 1/2, respectively. Exchange 1 values time one consumption more than exchange
2. In autarky, q 1 = (1 − β̄)π1 , q 2 = β̄ and q 3 = (π1 /2, π2 /2).
The restriction (8) implies that q 1 > q 2 , i.e. there exist profit opportunities for
arbitrageurs, buying on exchange 2 and delivering to exchange 1. It is also easy
to check that p3 = p∗ , the complete markets Walrasian state-price deflator. While
there may be many arbitrage opportunities, including some involving exchange 3,
the parameters chosen ensure that p2 < p3 < p1 , so that the main opportunity lies
in going long on exchange 2 and going short on exchange 1, while using exchange 3
merely to lay off the excess consumption in state 2 that results from this arbitrage
trade. Exchange 3 is not principally used because of its mispriced securities, but
serves as a passive financing exchange.
Under the optimal security design dk = pk − p∗ , or equivalently a complete set of
Arrow securities on each exchange, the arbitrageurs’ shadow valuation is p∗ , which
coincides with exchange 3’s valuation. As a result, there is no trade on 3. Arbitrageurs simply buy on 2 and deliver to 1, no longer relying on 3 for financing. On
19
the other hand, under the given initial asset structure, the arbitrageurs’ shadow val3
13
uation pA satisfies p3 −pA = p3M 3 −pA
Hence
M 3 6= 0, provided β is sufficiently small.
there is trade on exchange 3 prior to the innovation, and welfare is higher on this
exchange than under the optimal security design.
k
The example shows that not every exchange may benefit when all exchanges are
completed because the completion of markets may erode the advantages an exchange
may have had before the completion. Exchange 3 plays a valuable role in the arbitrage process at the initial asset structure, facilitating trade between the other
two exchanges. At a socially optimal asset structure, however, it becomes entirely
superfluous. This is reminiscent of what was also found in Willen (2005). We can
similarly show that when no clientele initially enjoys a trading advantage, then all
clienteles benefit from optimal security design.
Proposition 6.3 (Pareto-improving security design) Consider a clientele economy, with an initial asset structure that satisfies S. Then no clientele can be worse
off at a socially optimal asset structure.
Condition S(b) means that the initial asset structure is already socially optimal. In
the case of common payoff matrices, condition S(a), no exchange is at a trading
advantage at the initial equilibrium as trades that can be executed on one exchange
can equally be carried out on some other exchange.
Note that, in going from an initial asset structure to a socially optimal one,
we allow for the possibility of removing some of the initial assets. When we restrict
attention to innovation (not necessarily optimal) of additional assets, Proposition 6.3
extends to the general case where agents may be heterogeneous within exchanges.
Proposition 6.4 (Pareto-improving innovation) Suppose arbitrageurs introduce
new assets, and S is satisfied at both the initial and the post-innovation asset structure. Then no investor can be worse off after the innovation.
In particular, starting from a common asset structure dk = d, all k, if arbitrageurs
innovate optimally (i.e. the post-innovation asset structure satisfies S(b)), all agents
are better off. Even though arbitrageurs are solely interested in their own profits, in
this case maximal profit extraction means providing each investor with his favorite
assets.
The intuition for Proposition 6.4 is clearest in the limiting equilibrium as N goes
to infinity. We know from Proposition 4.1 that the complete-markets Walrasian
state-price deflator applies in this equilibrium. With quadratic preferences, adding
13
A straightforward but tedious derivation shows that pA (1)
and pA (2)
√
2
2β̄ +β̄+
=
1
1 2
1
2 −λ (3/2−β̄)+(λ ) π2 (1−β̄)
1−2λ1 +(λ1 )2 π2
=
1
1 2
1
2 −λ +(λ ) π2 (1−β̄)
1−2λ1 +(λ1 )2 π2
, both of which are nonnegative provided β 3
4β̄ 4 −(4+8π1 )β̄ 3 +8π1 β̄ 2 +2β̄
≤
. The deflator pA can be obtained more easily by exploiting the
general results in Rahi and Zigrand (2007b).
4(π1 −β̄(1+π1 ))
20
assets leaves the state-price deflator unchanged. Innovation cannot hurt investors as
it does not affect the prices of the initial assets. It should be noted, however, that we
do require that both the pre- and post-innovation asset structures satisfy S. This is
the case, for example, when agents have access to the same asset markets before and
after the innovation. In Example 3, S was not satisfied at the initial equilibrium,
and we saw that completing markets on all exchanges moved prices unfavorably for
some agents.
These results should be contrasted with those in Willen (2005) and Acharya and
Bisin (2006). These papers use a CARA-Gaussian setting, in which introducing
a new asset does not affect the prices of the risky assets that are already being
traded. Adding an asset is therefore always welfare-improving, provided there is no
consumption at the initial date. With consumption at both dates, and with the riskfree asset available for trade, adding an asset increases the risk-free interest rate,
so that agents who are net borrowers may be worse off. This is the only channel
through which financial innovation can hurt agents. We should also point out that
some of the welfare results in Willen (2005) and all of those in Acharya and Bisin
(2006) use a welfare measure involving lumpsum transfers, while we follow the more
standard approach of using the Pareto criterion.
Proposition 6.1 does not extend to the heterogeneous agent case. The minimal
equilibrium security design {pk −p∗ } will in general be Pareto dominated by complete
markets on every exchange. In a clientele economy both these asset structures are
socially optimal, and indeed payoff-equivalent for every investor and arbitrageur.
This is not so with heterogeneous agents. If there is sufficient heterogeneity, with
I k > S distinct investors on each exchange k, there will typically be S linearly
independent optimal net trades on every exchange, so that the only socially optimal
asset structure is the the one with complete markets on all exchanges.
The reason why the security design {pk −p∗ } fails to be socially optimal is because
arbitrageurs only care about aggregate valuations on the various exchanges. They do
not consider the effects of their security choice on the intra-exchange reallocation of
resources that occurs when investors on a given exchange trade the security among
themselves. Such intra-exchange trades are absent in a clientele economy (pk,i = pk ,
all k), which is why profit maximization by arbitrageurs is socially efficient in that
case.
7
Macro Markets
Robert Shiller (see Shiller (1993)) has argued that one of the most important categories of missing markets are markets for country GDP, which he calls “macro markets.” Stock markets for instance allow investors to trade only the small component
of national income that corresponds to corporate profits.
Since each trade needs a counterparty, macro markets can only be successful if
there is both a demand and a supply for a claim on each country’s GDP stream,
not least because otherwise innovators would not choose to launch such products.
21
Demand and supply must naturally be international. In that spirit, we proceed to
study macro markets within our formal model, reinterpreting exchanges as countries.
Assume initially that there are two countries, k = 1, 2. We have seen that
arbitrageurs would find it optimal to introduce one single (further) asset in country
k, pk − p∗ , which in this simple example means a payoff collinear with (β 1 ω 1 − β 2 ω 2 ).
Since ω k amounts in fact to GDP in country k, the equilibrium security corresponds
to the depth-weighted difference of the GDPs in the two countries.
An asset structure d1 = d2 = (ω 1 , ω 2 ) would also be an equilibrium of the design
game, albeit not a minimal one. What does come out of this, though, is that introducing country k’s GDP as a tradable asset in country k only is optimal neither for
arbitrageurs nor for investors. Investors in country k want to be able to simultaneously sell a portion of their GDP and diversify their portfolios by buying a fraction
of world GDP.
With K arbitrary, arbitrageurs could therefore introduce claims to the GDP of
each and every country in all countries. But it is more reasonable and cheaper to
introduce only the ideal security in country k, which is the difference of the GDP
outcomes of country k with the rest of the world, properly weighted by depth.
With this caveat, Shiller’s conjecture as to the importance of GDP markets is
mirrored in our model: those are indeed the equilibrium assets designed and traded
by profit-maximizing innovative institutions. Nevertheless, while macro GDP markets are optimal for arbitrageurs, Proposition 6.2 should caution us as to their social
welfare properties. Shiller has not specified any particular social welfare function,
and therefore claims must be evaluated carefully. Even with homogeneous investors
within each country, some countries may lose out from the introduction of macro markets, especially those that provide substitute insurance. Realistically, these countries
may also be the ones with the greatest political influence to prevent macro markets.
Finally, while Shiller’s conjecture does yield a social optimum with homogeneous
investors within each country, this is no longer true when they are heterogeneous.
Even though macro markets are the ones that innovators will in fact establish, on
their own they are not socially efficient in general.
8
Relationship to the Literature
This paper lies at the intersection of two distinct literatures—the literature on security design and that on segmented markets. Research on security design in an incomplete markets framework is surveyed in Allen and Gale (1994) and Duffie and Rahi
(1995). A number of papers study security design in a two-period mean-variance
setting. Of particular relevance to the present paper are Demange and Laroque
(1995), Rahi (1995), Willen (2005), and Acharya and Bisin (2006).14 The canonical
14
All of these papers assume CARA utility and normal distributions (Demange and Laroque
(1995) contains some general results as well). We do not follow this route since nonnegativity
restrictions on arbitrageurs’ final wealth are a key feature of our analysis and this makes the use of
normal distributions problematic. Willen (2005) and Acharya and Bisin (2006) are also discussed
22
problem posed in these papers is that of characterizing the optimal set of assets,
given an exogenous bound on the number of assets. The optimal assets are maximal
eigenvectors of a suitably defined matrix that depends on the endowments of all the
agents. Similar results are obtained in the literature on exchange-driven innovation
(e.g. Duffie and Jackson (1989) and Cuny (1993)). This class of security design results is very different from those that we report in this paper. In particular, these
models cannot account for the innovation of redundant assets.
In a model of securitization of an issuer’s assets, in which investors face short
sales restrictions, Allen and Gale (1988) obtain a security design that is superficially similar to ours in that the securities innovated are redundant and marketed to
various clienteles. In other respects, however, their characterization of “extremal”
securities is completely different from ours, and is not generated by issuers catering
to unspanned risks. Indeed, the economy-wide asset structure is exogenously given
in their model and the securities issued are redundant by assumption.
DeMarzo and Duffie (1999) and DeMarzo (2005) offer an explanation of the pooling and tranching of securities in a setup with risk neutral agents and privately informed issuers. Our framework, while not specifically designed to study asset-backed
securitization, delivers a complementary explanation, driven by market incompleteness and participation constraints, of intermediaries buying securities from some
investors, and then using these as “collateral” to sell securities to various clienteles.
The segmented markets framework that we employ in this paper is adapted from
Zigrand (2004, 2006) where the asset structure is taken to be exogenous. That not all
asset prices are determined simultaneously by the same marginal perfectly diversified
investor would be considered an axiom by practitioners. It has also been the basis
of early academic research, particularly in fixed income—see the market segmentation hypothesis posited by Culbertson (1957) or the preferred habitat hypothesis of
Modigliani and Sutch (1966). However, it is only recently that a series of papers have
tried to empirically quantify the extent to which state prices differ across markets.
One of the first systematic studies is the paper by Chen and Knez (1995). They
find that the NYSE and NASDAQ are priced by sets of state prices which are close
in mean-squared distance but do not intersect, showing that the marginal investors
on the two markets are close but distinct. There is also a prolific literature on the
international APT as well as on home bias, wherein a national stock market is held
and priced predominantly by national investors (see Lewis (1999) for a survey). More
recently, there have been a number of event studies of changes in the composition of
the S&P 500 index (see for instance Massa et al. (2005)). Around the time of the
addition a stock to the index, mutual funds benchmarked to the index have an incentive to purchase the stock, and their marginal valuations are shown to differ from
those of the market at large. Part of the resulting arbitrage is in fact performed by
the managers of the company admitted to the index. Similarly, Da and Gao (2006)
provide empirical evidence supporting the view that a sharp rise in a firm’s default
likelihood causes a change in its shareholder clientele as mutual funds decrease their
in Section 6.
23
holdings of the firm’s shares. This liquidity shock is initially absorbed by marketmakers before large traders move in to provide the liquidity. Gabaix et al. (2007),
in a study of the mortgage-backed securities (MBS) market, show that collateralized
mortgage obligations (CMOs) are priced not by the marginal investor of the broader
market whose state prices depend on the aggregate wealth of the economy, but by
investors wholly specialized in the MBS market. The intermediaries who purchase
the mortgages and transform them into CMOs play the role of our arbitrageurs. Finally, Blackburn et al. (2006) find that the marginal investors in options on growth
and value indices exhibit different degrees of risk aversion, growth investors being
less risk averse.
9
Conclusion
In this paper we analyze what happens if securities are designed not by a benevolent
social planner, or a derivatives exchange, or by companies issuing financial assets
based on the hitherto nontraded cash-flows of closely-held real assets, but by large
traders such as investment banks and hedge funds. We believe this corresponds
closely to what happens in actual markets, with asset innovations completing markets for certain groups of investors (“exchanges”) rather than for the economy as a
whole. For instance, capital-protected investment vehicles have recently stood in the
spotlight again, despite the fact that such assets are largely redundant. Price-fixing
retail banks sell them at a markup to their clients, and investment banks sell them
at a markup to the retail banks. Investment banks in turn have the ability to hedge
them in the underlying markets.
We answer the question as to which assets we should expect to see in an economy
with a variety of exchanges. Interestingly, we are able to provide an explicit and
minimal answer: an exchange is offered to trade the difference between its own
state-price deflator and the depth-weighted economy-wide state-price deflator. Only
one asset is introduced per exchange.
Depth plays a major role in this paper. If introducing a new security was costly
in our model, we would see that everything else equal, the only exchanges on which
innovation occurs are the deep exchanges. Shallow exchanges would be innovated
upon if the gains from trade are large enough to compensate for their shallowness.
One consequence is the adage, well-known to practitioners, that derivative securities
can only be successful if there is sufficient demand for them from some clientele, i.e.
from the end-users. This is evident in our model: no matter how many intermediaries
trade on an exchange, if the depth of the end-users tends to zero (for instance if the
number of investors tends to zero), trading on the exchange vanishes as well.
We are not interested in just the equilibrium asset structure. The general equilibrium nature of our setup also allows us to study welfare properties. It is far from
obvious if the securities introduced by arbitrageurs in order to extract the largest
profits from wedges in investors’ marginal willingness to pay lead to a socially optimal outcome. Arbitrageurs are driven by mispricings and depth considerations, not
24
by socially beneficial gains from trade. Still, we can show that if investors within an
exchange have identical valuations, so they form a homogeneous clientele, then an
equilibrium structure is socially optimal and independent of the degree of competition between arbitrageurs. However, equilibrium allocations are not socially optimal
and do depend on the degree of competition. We also provide a necessary and sufficient criterion that characterizes those investors who gain and those who lose from
said innovations. If investors within an exchange are heterogeneous, the equilibrium
security design is no longer socially optimal. The reason is that arbitrageurs ignore
the gains from trade within exchanges since they only profit from inter-exchange
reallocations of resources. The equilibrium asset structure is therefore geared toward extracting the maximum inter-exchange gains from trade, at the expense of
intra-exchange gains from trade.
Our paper provides a rich framework for studying a number of issues, some of
which we have only barely touched on here. One is a more general concept of liquidity.
Arbitrageurs provide liquidity by mediating trades, much in the same way as marketmakers do. This suggests a measure of liquidity that ties together the seemingly
disparate notions of depth, bid-ask spreads, trading volume, and gains from trade.
We pursue this idea in Rahi and Zigrand (2007c). Second, the arbitraging scenario
we have studied in this paper, wherein all arbitrageurs are simultaneously active
on all exchanges, is but one possible description of intermediation in a segmented
economy. In Rahi and Zigrand (2007a) we look at the case where arbitrageurs choose
a subset of exchanges on which to trade, and analyze the resulting distribution of
arbitrageur activity. We are able to derive simple rules as to the optimal exchanges
on which to innovate. Quite intuitively, arbitrageurs gravitate to those exchanges
which, other things being equal, are deeper and stand to gain most from trading with
other exchanges. We also show, for a class of economies, that a hub-spoke network,
in which all arbitrageur activity is channeled through a central exchange (the “hub”),
leads to higher arbitrageur profits than any other network structure.
25
A
Appendix: Proofs
In the Appendix we adopt matrix notation that simplifies the proofs considerably.
Rather than viewing asset payoffs on exchange k as random variables dk , we stack
them into an S × J k matrix Rk . The j’th column of Rk corresponds to the j’th asset,
listing its payoffs in each state s ∈ S. In this notation the set of marketable payoffs
M k is the column space of Rk , denoted by hRk i.
Let πs be the probability of state s, and denote by Π the S × S diagonal matrix
with πs at position (s, s). A state-price deflator for (q k , Rk ) is a vector p ∈ RS such
>
that q k = Rk p.15 In other words, it is convenient to think of state-price deflators as
vectors instead of random variables. Similarly, the expression E[xy] can be written
as x> Πy, where the random variables x and y are viewed as vectors in RS . In our
finite-dimensional setting, the inner product space L2 is the space RS endowed with
the inner product (x|y) = x> Πy. Then xM k = P k x, where P k is the orthogonal
projection operator in L2 onto hRk i, given by the idempotent matrix
>
>
P k := Rk (Rk ΠRk )−1 Rk Π.
(9)
An explicit derivation of P k can be found in the proof of Lemma 3.2 below. The
1
notation k · k2 stands for the L2 -norm defined by kxk2 := (x> Πx) 2 , for x ∈ RS .
Proof of Lemma 3.1 Investor (k, i)’s utility can be written as
U k,i = ω0k,i − q k · θk,i + 1> Π(ω k,i + Rk θk,i ) −
β k,i k,i
(ω + Rk θk,i )> Π(ω k,i + Rk θk,i ), (10)
2
where 1 is the S × 1 vector of ones. The first order condition for the investor’s
optimization problem is
>
>
−q k + Rk Π1 − β k,i Rk Π(ω k,i + Rk θk,i ) = 0,
(11)
which gives us his asset demand function (recall that pk,i := 1 − β k,i ω k,i ):
1
>
>
(Rk ΠRk )−1 [Rk Πpk,i − q k ].
k,i
β
P
We can now use the market-clearing condition i∈I k θk,i (q k ) = y k to deduce:
θk,i (q k ) =
>
q k (y k ) = Rk Π[pk − β k Rk y k ].
(12)
(13)
>
Thus q k (y k ) = Rk Πp̂k , where p̂k = pk − β k Rk y k .
15
The symbol > denotes ”transpose.” We adopt the convention of taking all vectors to be column
vectors by default, unless transposed.
26
Proof of Lemma 3.2 Using (13), we can write the Lagrangian for arbitrageur n
as follows:
X
X
>
L=
[pk − β k Rk y k,n − β k Rk y k,\n ]> ΠRk y k,n − pA,n Π
Rk y k,n ,
k
k
where y k,\n is the aggregate supply of assets on exchange k of all arbitrageurs but
n. pA,n is the Lagrange multiplier vector attached to the no-default constraints, and
can be interpreted as a (shadow) state-price deflator of the arbitrageur. The first
order conditions are:
>
Rk Π[pk − β k Rk y k,\n − 2β k Rk y k,n − pA,n ] = 0,
k∈K
(14)
together with complementary slackness:
"
pA,n ≥ 0,
X
Rk y k,n ≤ 0,
and pA,n ·
k
#
X
Rk y k,n = 0.
(15)
k
The existence of the multipliers follows as usual from the linearity of the inequalities,
as shown in Arrow et al. (1961).
We first demonstrate that a CWE is symmetric, i.e. y k,n does not depend on n,
and we can choose pA,n = pA for all n. The reaction correspondence of arbitrageur
n, for given supply of the remaining arbitrageurs {y k,\n }, is single-valued due to the
strict concavity of the program. From the first order conditions (14) and (15),
y k,n =
1
>
>
(Rk ΠRk )−1 Rk Π[pk − pA,n ] − y k ,
k
β
and
pA,n ≥ 0,
k∈K
X 1
X
k k
A,n
Rk y k ≤ 0
P
(p
−
p
)
−
k
β
k
k
(16)
(17)
with complementary slackness state-by-state. In other words, there is some pA,n ≥ 0
satisfying the no-default and complementary slackness conditions so that the unique
y k,n chosen if all others choose y k,\n is given by (16). Many feasible pA,n may exist,
>
but any pA,n and p̃A,n that represent the reaction function must satisfy Rk ΠpA,n =
>
Rk Πp̃A,n , otherwise single-valuedness is violated.
It follows that y k,n cannot depend on n at an equilibrium. Indeed, assume to the
0
contrary that an equilibrium {y k,n } is such that y k,n 6= y k,n for some k and some pair
0
(n, n0 ). Then (16) implies that pA,n 6= pA,n . The inequalities (17) for arbitrageurs n
0
and n0 depend only on the aggregate quantities y k , for all k. So given {y k,n }, pA,n
0
>
>
is also a valid shadow price for arbitrageur n. But then Rk ΠpA,n = Rk ΠpA,n ,
0
implying that y k,n = y k,n .
Having established symmetry, we can easily solve (16) for y k,n and verify that
Rk y k,n =
1
· P k (pk − pA ),
k
(1 + N )β
27
k∈K
(18)
where P k is given by (9). This is in fact the unique solution. From (15) and (18),
the complementary slackness conditions can be written as pA ≥ 0 and
X 1
P k (pk − pA ) ≤ 0,
(19)
k
β
#
" k
X 1
P k (pk − pA ) = 0.
(20)
pA ·
k
β
k
Note that pA can be chosen not to depend on N . We will need these conditions later.
Proof of Lemma 3.3 Using (13) and (18),
N
k
k>
k
k k
A
q =R Π p −
P (p − p )
1+N
N
k
A
k>
k
(p − p )
=R Π p −
1+N
1
N
k>
k
A
=R Π
p +
p .
1+N
1+N
(21)
Proof of Lemma 3.4 Using (18), (21) and (20), in that order, some straightforward
algebra gives us the equilibrium profits of arbitrageur n, for asset structure {Rk }:
X
Φ({Rk }) =
q k · y k,n
k
X 1 >
1
k
k
k
A
·
p
ΠP
p
−
p
(1 + N )2 k β k
X 1
1
>
=
·
(pk − pA ) ΠP k pk − pA
2
k
(1 + N )
β
k
X 1
1
·
kP k (pk − pA )k22 .
=
(1 + N )2 k β k
=
(22)
Proof of Lemma 4.1 From (12), setting Rk = I, the portfolio of agent (k, i) at
1
p∗ is θk,i = β k,i
(pk,i − p∗ ). Therefore the aggregate portfolio on exchange k is θk :=
P
P
k,i
= β1k (pk −p∗ ). Now the Walrasian market-clearing condition, k∈K θk = 0,
i∈I k θ
P
P
gives us p∗ = k∈K λk pk . Moreover, k∈K λk pk = 1 − βω, which is nonnegative by
assumption.
P k k
∗
Proof
of
Lemma
4.2
Recall
that
p
=
Under S(a),
k λ p by Lemma
P k k k
P k k
P 4.1.
∗
1
∗
k k k
∗
k λ P (p − p ) = P
k λ (p − p ) = 0. Under S(b),
k λ P (p − p ) =
28
P
λk (pk − p∗ ) = 0. Thus k λk P k (pk − p∗ ) = 0 under S. It follows that pA = p∗
satisfies the complementary slackness conditions (19) and (20). Furthermore, p∗ ≥ 0
by Lemma 4.1. Therefore, p∗ is a valid Lagrange multiplier for the arbitrageur’s
optimization problem.
P
k
In order to prove Proposition 5.1, we need to establish the following result.
Fact A.1 kP k vk2 ≤ kvk2 , for all v ∈ RS . Moreover, kP k vk2 = kvk2 if and only if
P k v = v.
Proof :
>
>
kP k vk22 = v > ΠRk (Rk ΠRk )−1 Rk Πv
>
>
= (Π1/2 v)> Π1/2 Rk (Rk ΠRk )−1 Rk Π1/2 (Π1/2 v)
>
>
Defining x := Π1/2 v and y := Π1/2 P k v = Π1/2 Rk (Rk ΠRk )−1 Rk Π1/2 x, we see that
= x · y = kxkkyk cos(θ), where θ is the angle between x and y, and where
kP k vk22 √
kxk := x · x. Now y · y = x · y ≥ 0. If x · y = 0, the result follows. Otherwise we
√
get kP k vk22 = x · y = kxk x · y cos(θ) which we can solve for x · y = kxk2 cos2 (θ).
We therefore find that kP k vk22 = kxk2 cos2 (θ) ≤ kxk2 = v > Πv = kvk22 .
Now suppose that kP k vk2 = kvk2 . We want to show that P k v = v. If x · y = 0,
then x · y = kP k vk22 = kvk22 = 0, so that P k v = v = 0. If, on the other hand
x · y > 0, then kP k vk2 = kvk2 implies that cos2 (θ) = 1, i.e. x and y are collinear.
But x · y = y · y 6= 0. Hence x = y, or P k v = v.
Proof of Proposition 5.1 Proof of 1 ⇒ 2: Suppose pk − p∗ ∈ hRk i, k ∈ K.
Then condition S holds so that, from Lemma 4.2, we can choose pA = p∗ . From (22),
noting that P k (pk − p∗ ) = pk − p∗ , equilibrium arbitrageur profits are given by
Φ=
X 1
1
·
kpk − p∗ k22 .
(1 + N )2 k β k
(23)
In order to establish that the proposed asset structure is optimal for every arbitrageur, we need to show that
X
X
λk kpk − p∗ k22 ≥
λk kP k (pk − pA )k22 .
(24)
k
Fact A.1 implies that
X
k
k
λk kpk − p∗ k22 ≥
X
k
29
λk kP k (pk − p∗ )k22 .
(25)
Furthermore, noting that x> Ax − y > Ay = (x − y)> A(x + y), for any vectors x, y,
and any square matrix A, we have
X
X
λk kP k (pk − p∗ )k22 −
λk kP k (pk − pA )k22
k
k
=
X
=
X
=
X
=
X
λ [(p − p ) ΠP (pk − p∗ ) − (pk − pA )> ΠP k (pk − pA )]
k
∗ >
k
k
k
λk (pA − p∗ )> ΠP k (2pk − p∗ − pA )
k
λk (pA − p∗ )> ΠP k (pA − p∗ ) + 2(pA − p∗ )> Π
X
k
λk P k (pk − pA )
k
k
k
A
λ kP (p − p
∗
)k22
∗>
− 2p Π
X
k
k
k
k
A
λ P (p − p )
(using (20))
k
≥0
where the last inequality follows from (19) and the fact that p∗ ≥ 0. In conjunction
with (25), this implies (24).
Proof of 2 ⇒ 3: Suppose an optimal asset structure {Rk } is not a Nash equilibrium asset structure, i.e. an arbitrageur can deviate such that the resulting asset
structure increases his payoff. Then the payoff of all arbitrageurs is higher, contradicting the optimality of the initial asset structure {Rk }.
Proof of 3 ⇒ 1: Suppose a Nash equilibrium asset structure {Rk } does not satisfy
the condition pk − p∗ ∈ hRk i, k ∈ K. Then (25) holds with strict inequality due
to Fact A.1, implying that profits are strictly lower than (23). Thus an arbitrageur
can strictly increase his payoff by introducing the asset pk − p∗ on each exchange k
where it is not already available. This contradicts the supposition that {Rk } is a
Nash equilibrium asset structure.
Proof of Lemma 6.1 Using investor (k, i)’s first order condition (11), we can
write his utility (10) as:
U k,i = ω0k,i + 1> Πω k,i −
β k,i k,i > k,i β k,i k k,i 2
ω Πω +
kR θ k2 .
2
2
Note that U k,i depends on the asset structure only through the term kRk θk,i k22 . From
(12) and (3), we see that
1 k k,i
P (p − p̂k )
β k,i
1 k
N
k,i
k
k
A
= k,i P (p − p ) +
(p − p ) ,
β
1+N
Rk θk,i =
so that
W
k,i
:= β
k,i
kRk θk,i k22
2
1 N
k
k,i
k
k
A
= k,i P (p − p ) +
(p − p ) .
β
1+N
2
30
(26)
Proof of Proposition 6.3 Suppose the initial asset structure {Rk } satisfies S.
Then, from Lemma 4.2, we can choose pA = p∗ . Therefore,
kP k (pk − pA )k2 = kP k (pk − p∗ )k2 ≤ kpk − p∗ k2 ,
using Fact A.1. The result now follows from Proposition 6.2.
Proof of Proposition 6.4 Let the pre- and post-innovation asset structures be,
respectively, {Rk } and {R̄k }, with projection matrices {P k } and {P̄ k }. Since hRk i ⊂
hR̄k i, we have P k P̄ k = P k . Moreover, both asset structures satisfy S, so that we can
choose pA = p∗ in both cases, from Lemma 4.2. Let
ζ k,i := (pk,i − pk ) +
N
(pk − pA ).
1+N
We have kP k ζ k,i k2 = kP k P̄ k ζ k,i k2 ≤ kP̄ k ζ k,i k2 , using Fact A.1. The result now
follows from Lemma 6.1.
31
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